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Chapter 5 / Monopoly

5.3 Price Discrimination

In the monopoly model examined so far in this chapter, the price charged by the monopolist is linear and the same for everyone: each unit of the good is sold at the same price, regardless of how many units are purchased and of who the buyer is. In this section we extend the analysis by assuming that the monopolist can charge different prices for the same good, depending on the quantity purchased or on the consumer’s characteristics. As we will see, this practice, known as price discrimination, allows the monopolist to capture a larger share of the surplus generated by trade.

Economists traditionally distinguish This classification was introduced by the economist Arthur Cecil Pigou in 1920 in The Economics of Welfare. In the original version, second-degree discrimination was not exactly a nonlinear price, but rather an approximation to first-degree discrimination obtained by applying a limited number of possible unit prices. three forms of discrimination:


Perfect Price Discrimination

When the monopolist charges linear and uniform prices, each unit is sold at the same price: each consumer buys one more unit only if their willingness to pay for that unit is at least equal to that price. The monopolist could attract new customers and sell more to existing ones, but to do so would have to lower the price, reducing revenue on the units already sold: this is the price effect mechanism discussed earlier. The monopolist prefers to keep a higher price, thereby foregoing additional units that consumers would be willing to pay more than marginal cost for.

With perfect discrimination, the monopolist completely overcomes this constraint: each unit is sold at a price equal to the maximum willingness to pay for that unit — the highest price that some consumer is willing to pay for it — without having to lower the price of previous units. But if the price effect is zero, the marginal revenue function coincides with the market demand curve. The firm appropriates the entire surplus and is therefore incentivized to maximize it: the traded quantity is the competitive one, and deadweight loss is zero.

The next figure shows this result, assuming for simplicity that there are only two consumers and, for clarity, that the good is indivisible, such as subway tickets.

FIGURE 5.6

In the case of a divisible good, the reasoning is the same: every small fraction is sold at the price corresponding to willingness to pay, as indicated by the demand curve. The next figure illustrates the case of a mobile operator selling gigabytes of mobile data. We assume that the monopolist’s marginal cost is $MC=2$ and that market demand is $P = 10 - Q/50$, where $P$ is the price per gigabyte and $Q$ is the quantity (in millions of) gigabytes.

Two-Part Tariffs and Bundling

Perfect price discrimination may seem hard to implement in practice — and indeed it is. One possible concern is that, to implement it, the monopolist would appear to have to charge a different price depending on how many units a consumer has already purchased. In Figure 5.6, for example, Alice pays 8 euros for a ticket if she has not already bought any, 7 if she has already bought one, and so on. However, this is not the real reason why perfect discrimination is difficult to implement (we will discuss the real reasons below). There are in fact two strategies, both quite common in reality, that replicate the outcome of perfect discrimination: the two-part tariff and bundling.

A two-part tariff has two components:

The profit-maximizing two-part tariff is based on a very simple idea: choose the unit price so as to induce the consumer to buy all units that create surplus, and choose the fixed part so as to capture all of that surplus. To do this, the monopolist sets the unit price equal to marginal cost, so that the consumer is incentivized to buy every unit for which willingness to pay exceeds cost. At that point, the monopolist chooses an access fee equal to the net benefit the consumer would obtain by paying only the unit price. The consumer is willing to pay that fixed fee because they obtain the quantity they want at a still-convenient overall price; but in the end their surplus is driven to zero, and all of the value generated by trade goes to the firm.

Let’s return to the example of Figure 5.6, where each unit represents a subway ticket. Alice is willing to pay 9 euros for the first ticket, 7 for the second, and so on. Bruno is willing to pay 6 euros for the first ticket, 4 for the second, and so on. If the unit price is set at marginal cost (1.50 euros), Alice therefore buys 4 tickets, while Bruno buys 3. The surplus Alice would obtain by paying only the unit price is

The reasoning is unchanged if “Alice” and “Bruno” are not single consumers but, more realistically, homogeneous groups of consumers made up of many individuals with the same individual demand. The monopolist charges a fixed fee of 18 euros to each “Alice-type” consumer, a fixed fee of 7.50 euros to each “Bruno-type” consumer, and 1.50 euros per ticket to everyone. Quantities, revenues, and profits are simply scaled by the sizes of the two groups, without changing the logic or the results shown in the text.

\(\begin{gathered} (9 - 1.50) + (7 - 1.50) + \dots + (3 - 1.50) = 18 \end{gathered}\)

Similarly, the surplus Bruno would obtain is

\(\begin{gathered} (6 - 1.50) + (4 - 1.50) + (2 - 1.50) = 7.50 \end{gathered}\)

The monopolist can therefore charge Alice an access fee of 18 euros and Bruno a fee of 7.50 euros, in addition to the payment of 1.50 euros for each ticket. Total profit will be $18+7.50=25.50$ euros, as under perfect discrimination.

Now consider the example in Figure 5.7, where the good is mobile data. In that example marginal cost is $MC=2$ and market demand is $Q = 500 - 50P$. We can think of the latter as the aggregate demand of 100 identical consumers, each with individual demand $Q = 5 - P/2$. If the unit price is set at 2 euros, each consumer buys $4$ gigabytes, obtaining a surplus of 16 euros. The Car Sharing Two-part tariffs and bundling are widely used. Mobile operators (as in Figure 5.7) offer two-part tariffs in the form of monthly plans with a fixed fee and a per-GB or per-minute charge, or prepaid bundles that include a fixed amount of data and minutes at a single price. Public transport systems (as in Figure 5.6) often offer both prepaid carnets, e.g., of 10 rides (bundling), and subscriptions with a fixed fee and a unit price for additional rides (two-part tariff). Car-sharing services use similar formulas: on the one hand, monthly subscriptions plus a price per minute or kilometer; on the other hand, hourly or daily packages at a flat rate. monopolist can then offer a tariff with a unit price of 2 euros and an access fee of 16 euros. Total profit is once again equal to the maximum total surplus, 1,600 euros.

Bundling consists of offering the consumer a block of units at a fixed total price. The consumer can choose to buy the entire bundle or nothing, but cannot buy intermediate quantities.

Returning to the subway ticket example (Figure 5.6), we know that Alice is willing to pay a total of 24 euros for 4 tickets, and Bruno is willing to pay 12 euros for 3 tickets. The monopolist can simply offer Alice a 4-ticket bundle at a price of 24 euros and Bruno a 3-ticket bundle at a price of 12 euros. The monopolist’s revenue is 36 euros; cost is $7 \times 1.50 = 10.50$ euros; therefore profit is 25.50 euros: the same outcome as with the two-part tariff.

In the mobile data case (Figure 5.7), the consumer would buy 4 gigabytes at a price of 2 euros/GB, and the total value they assign to a 4 GB bundle is 24 euros. The monopolist can therefore sell a 4 GB bundle for 24 euros. Cost is 8 euros; profit is 16 euros per consumer, that is, 1,600 in total: once again, the same as with the two-part tariff.

Informational and Institutional/Economic Difficulties

Perfect price discrimination is a useful theoretical benchmark: it represents the limiting case in which the monopolist can extract all surplus by offering personalized nonlinear tariffs (for example, two-part tariffs or bundles). However, it is difficult to implement. The monopolist faces (at least) two major challenges:

Fidelity Cards Loyalty cards, apps, and online registrations are also used to collect data on purchasing habits, so as to better estimate demand for different customer segments. Some firms go further, using digital profiling to build personalized offers.

When consumers are identical (as we can think is the case in the example in Figure 5.7, or if in Figure 5.6 Bruno did not exist and consumers were many “Alices”), the first challenge is not a major problem. If offering nonlinear tariffs is neither prohibited nor useless, the monopolist can implement perfect discrimination by offering a single two-part tariff or a single bundle. On the other hand, if nonlinear tariffs are inapplicable, the story ends here as well: the best the monopolist can do is offer a uniform linear price, as in the basic analysis of the previous sections.

In reality, however, individuals are heterogeneous. This potentially makes both challenges relevant and, consequently, second-degree discrimination (for the informational problem) and third-degree discrimination (when nonlinearity is prohibited or useless). As we will see, these two types of discrimination are the best available strategies to address, respectively, the two challenges. The monopolist will not be able to replicate perfect discrimination but will still obtain higher profit than under a uniform linear price.


**Second-Degree Price Discrimination

Suppose the monopolist can offer nonlinear tariffs but cannot treat consumers differently because it does not know “who is who.” In this case, the best thing to do is to offer a menu of options and let each consumer self-select by choosing the preferred option. The analysis of this type of discrimination is, in general, quite complicated and requires tools from information economics that we do not have. We will therefore limit ourselves to a simple case, which builds on the data from the example shown in Figure 5.6.

We thus have Alice, who is willing to pay 9 euros for the first ticket, 7 for the second, and so on, and Bruno, who is willing to pay 4 euros for the first ticket, 3 for the second, and so on. In Figure 5.6 we saw that, with a single linear price for everyone, the maximum profit the monopolist can obtain is 14 euros, whereas with perfect discrimination the maximum profit is 25.50 euros. How much profit can be obtained if the monopolist can offer nonlinear tariffs but cannot tell Alice from Bruno?

One possibility is to offer a 4-ticket bundle at 24 euros and a 3-ticket bundle at 12 euros, as under perfect discrimination. But this is not a good idea. Doing so would induce not only Bruno but also Alice to choose the 3-ticket bundle. In fact, Alice prefers to pay 12 euros for 3 tickets, which for her are worth $9+7+5=21$ euros, rather than pay 24 for 4. In the first case her surplus is $21-12=9$, in the second it is zero. The monopolist’s profit would therefore be very low: $12+12-2\times(3+3)=12$. To make Alice buy the 4-ticket bundle, which for her is worth $9+7+5+3=24$ euros, it would have to be priced at 15 euros. The monopolist’s profit would then be $12+15-2\times(3+4)=13$. But the monopolist can do much better than that.

To earn more profit, the monopolist must reduce the attractiveness of the bundle designed for Bruno, so as to be able to raise the price of the bundle designed for Alice. The procedure to find the right strategy is simple. Starting from the two bundles offered under perfect discrimination (3 tickets at 12 euros, 4 tickets at 24 euros), progressively reduce the quantity and price of the bundle intended for Bruno, and at the same time set the price of the bundle intended for Alice so that she chooses it, until finding the combination that delivers the highest profit. We now illustrate the procedure.

Suppose we offer a 2-ticket bundle, instead of 3, at 10 euros (what Bruno is willing to pay for two tickets). This option is now less attractive to Alice: if she chose it, she would obtain a surplus equal to the value to her of two tickets ($9+7=16$) minus $10$, i.e. $6$. To ensure that Alice buys the 4-ticket bundle, we can then set the price of that bundle so that Alice is left with at least a surplus of $6$. The value of 4 tickets for Alice is 24, so we can sell the 4-ticket bundle at $18$ euros. At this point Alice chooses “4 tickets at 18 euros” and Bruno chooses “2 tickets at 10 euros”. Revenue is $18+10=28$ while costs are $1.50\times(3+4)=10.50$. Profit is $28−10.50 = 17.50$.

Continuing the procedure, suppose we offer a 1-ticket bundle at 6 euros (what Bruno is willing to pay for one ticket). This option is even less attractive to Alice: by choosing it, she would obtain a surplus of $9-6=3$. To ensure that Alice buys the 4-ticket bundle, we can then set its price at $24-3=21$ euros. At this point Alice chooses “4 tickets at 21 euros,” Bruno chooses “1 ticket at 6 euros.” Revenue is $21+6=27$ and costs are $1.50\times 5=7.50$, so profit is $27−7.50 = 19.50$.

Is it possible to earn an even higher profit by further reducing the quantity intended for Bruno? That would mean reducing it to zero, i.e. offering only a 4-ticket bundle intended for Alice. With no alternative option to draw Alice away, the monopolist could set the bundle’s price at the maximum Alice is willing to pay, 24 euros. But profit would be lower: $24-1.50\times 4=18$. The best strategy is therefore the one described in the previous step of the procedure: single tickets at 6 euros each, and 4-ticket bundles at 21 euros. The monopolist applies a quantity discount, selling single tickets at 6 euros, and 4-ticket bundles at 5.25 euros per ticket.

Quantity discounts like the one we just saw—and like the “3×2,” “the cheapest at half price,” etc. that we often observe in real markets—are classic nonlinear prices. The seller applies a unit price that decreases with quantity, pushing customers with higher willingness to pay to buy more. Other examples include tiered plans (mobile, cloud), base/premium versions, “clip-out” coupons, and free-shipping thresholds in online shopping. In all these cases the firm, unable to distinguish consumer types, offers a menu that makes it optimal for each customer to choose the option “designed for them,” allowing the firm to extract more surplus than with a single linear price.


Discrimination Based on Observable Characteristics

Now suppose information is not a problem (the monopolist knows “who is who” and can treat consumers differently) but it cannot (or it is useless to) apply nonlinear prices. The best thing to do in this case is to offer personalized linear prices. As we will see, here too the monopolist will have to give up part of the surplus — this time because of the linear pricing constraint — but will still obtain higher profit than under a uniform linear price.

To understand how discrimination based on observable characteristics works, consider the example of Office suite licenses. Since almost all users purchase only a single license, it is natural to think in terms of linear prices: quantity discounts or other forms of nonlinear pricing would not increase profits. It is also plausible that professionals and firms have a higher willingness to pay compared to, for instance, students, and that the monopolist is able to distinguish between the two segments (for example, by verifying student status) and apply different prices.

Let us then suppose there are two groups of users: professionals/firms (group A), with higher willingness to pay, and students (group B), with lower willingness to pay. The following figure, where for simplicity — and to facilitate comparison with perfect discrimination — we use the same numbers already seen in Figure 5.6, shows how the monopolist can increase profits by charging a higher price to professionals and a lower price to students.

FIGURE 5.8

Let us now consider a divisible good, for example Discounts and reduced fares for young people and students are very common in railway transport in Europe and in many other countries, where age or student status grants access to lower prices. kilometers of travel on a railway line (measured in blocks of 100 km). Suppose the population consists of two groups of users: adults, whose aggregate demand curve is $Q_A=20-P$, and young people, whose demand curve is $Q_B=12-P$, where $Q_A$ and $Q_B$ denote the millions of hundreds of kilometers demanded at each given price. The marginal cost is $MC=2$ euros / 100km. If the monopolist applies a single price, it will have to choose a compromise that does not maximize profit in either group. If instead it segments the market by charging a lower price to young people and a higher one to adults, it obtains a higher profit. The following figure illustrates the result.


FIGURE 5.9

In the previous section we saw that the linear price set by a nondiscriminating monopolist is higher the less elastic demand is at the optimum point. The same reasoning applies to third-degree price discrimination: the Lerner rule holds separately for each group. In each If instead the monopolist does not discriminate, it chooses $P=9$, but this intermediate price does not maximize profits in either group. In fact, at $P=9$ we have \(-\frac{1}{E^D_A} > \frac{P - MC}{P} > -\frac{1}{E^D_B}\) The first inequality means that $MR_A< MC$, the second that $MR_B> MC$. In other words, starting from $P=9$ the monopolist has an incentive to reduce quantity (raise price) for group $A$ and to increase quantity (lower price) for group $B$. market segment the monopolist chooses the optimal prices $P_A=11$ and $P_B=7$ by imposing the conditions

\(\begin{gathered} \frac{P_A - MC}{P_A} = -\frac{1}{E^D_A} \qquad \frac{P_B - MC}{P_B} = -\frac{1}{E^D_B} \end{gathered}\)

where $E^D_A$ and $E^D_B$ are the demand elasticities in the two groups. In our example, adult demand is less elastic at a given price, and the monopolist therefore applies a higher price to this group. By contrast, student demand is more elastic, and the optimal price is lower. This logic explains why, in real markets, we regularly observe reduced fares for students or other categories with higher price sensitivity: these are not “altruistic” discounts, but rather a profit-maximizing strategy based on differences in demand elasticity.

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